Quadratic forms associated to stratifying systems · subcategories of modules. In the context of...

Journal of Algebra 302 (2006) 750–770

www.elsevier.com/locate/jalgebra

Quadratic forms associated to stratifying systems ✩

Eduardo Do N. Marcos a, Octavio Mendoza b,∗, Corina Sáenz c,Rita Zuazua d

a Departamento de Matemática, Universidade de São Paulo, Caixa Postal 66.281, São Paulo, SP 05315-970, Brazilb Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, C.P. 04510, México, DF, Mexico

c Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, Ciudad Universitaria,C.P. 04510, México, DF, Mexico

d Instituto de Matemáticas, Unidad Morelia, UNAM, A.P. 61-3 Xangari, C.P. 58089, Morelia Michoacán, Mexico

Received 27 October 2004

Available online 16 June 2006

Communicated by Kent R. Fuller

Dedicated to Professor Claus Michael Ringel on his 60th birthday

Abstract

Let R be an algebra, and let (θ,�) be a stratifying system of R-modules. If the category F(θ) isθ -directing, then we prove that indF(θ) is finite. In order to do that, we introduce a quadratic form qθwhich depends on θ . Moreover, we also give sufficient conditions to get the correspondence X �→ dim θXfrom indF(θ) to the set of positive roots of qθ .© 2006 Elsevier Inc. All rights reserved.

Keywords: Standardly stratified algebras; Quadratic forms; Stratifying systems

0. Introduction

In this paper, algebra means finite-dimensional basic algebra over an algebraically closedfield k. If R is an algebra, the category of finitely generated left R-modules is denoted by mod R,and the usual duality Homk(−, k) : mod R → mod Rop is denoted by D. All the subcategories of

✩ The authors thank the financial support received from Project PAPIIT-UNAM IN115905. The first author has aproductivity grant from CNPq, Brazil.

* Corresponding author.E-mail addresses: [email protected] (E.D.N. Marcos), [email protected] (O. Mendoza),

[email protected] (C. Sáenz), [email protected] (R. Zuazua).

0021-8693/$ – see front matter © 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2006.05.017

E.D.N. Marcos et al. / Journal of Algebra 302 (2006) 750–770 751

mod R to be considered are full subcategories. Given f :M → N and g :N → L morphisms inmod R we denote the composition of f and g by gf , which is a morphism from M to L.

Given a class C of R-modules, we denote by F(C) the subcategory of mod R whose objectsare the zero module and all modules which are filtered by modules in C. That is, a non-zeroR-module M belongs to F(C) if there is a finite chain

0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm = M

of submodules of M such that Mi/Mi−1 is isomorphic to a module in C for all i = 1,2, . . . ,m.In fact, F(C) is closed under extensions and F(∅) = {0}.

Let R be an algebra and {ε1, . . . , εs} be a complete set of primitive orthogonal idempotents.Then, we fix the natural order on the set of indices [1, s] = {1, . . . , s}. Let P(i) = Rεi be theindecomposable projective R-module corresponding to the idempotent εi , and S(i) be the sim-ple top P(i)/ radP(i) of P(i) for 1 � i � s. The standard module RΔ(i) is, by definition, themaximal factor module of P(i) without composition factors S(j) for j > i. We denote by RΔthe set {RΔ(i)}i∈[1,s].

We recall that an algebra R, with a fixed total order of the simple modules, is called stan-dardly stratified if RR ∈ F(RΔ). A standardly stratified algebra R is called quasi-hereditary ifdimk End(RΔ(i)) = 1 for any i. Quasi-hereditary algebras were introduced by E. Cline, B.J. Par-shall and L.L. Scott in [3] and standardly stratified algebras by V. Dlab in [5]; furthermore, a newtheory of stratified algebras has been given in [4].

Instead of a partial order on the iso-classes of simple modules, as was done in [3], we will con-sider only total orders. In addition, typical examples of quasi-hereditary algebras are hereditaryalgebras. In fact, it was proved by V. Dlab and C.M. Ringel in [6] that an algebra is hereditary ifand only if it is a quasi-hereditary algebra for any total order of the simple modules.

Quadratic forms play an important role in the representation theory of finite-dimensional alge-bras. Thus, it makes sense to try them also for “relative representation theory,” that is, for certainsubcategories of modules. In the context of quasi-hereditary algebras, S. Liu and C. Xi consid-ered in [9] hereditary algebras as quasi-hereditary algebras. They introduced a quadratic formq

RΔ for a quasi-hereditary algebra R, and proved that if R is a hereditary algebra, then F(RΔ)is finite if and only if q

RΔ is weakly positive. Later on, in [2], B. Deng studied the categoryF(RΔ) with R a quasi-hereditary algebra. Analogously as it is usually done for mod R, B. Dengstudied RΔ-directing and RΔ-omnipresent modules in F(RΔ), and proved that the existence ofa RΔ-directing and RΔ-omnipresent module in F(RΔ) implies that all standard modules haveprojective dimension at most 2. Afterwards, by using the process of standardization introducedby V. Dlab and C.M. Ringel in [7], B. Deng showed that the study of RΔ-directing modules inF(RΔ) can be reduced to the study of those over certain quasi-hereditary algebra B which admitsa BΔ-directing and BΔ-omnipresent module. Moreover, B. Deng proved that, for the algebras Rand B , the quadratic forms associated to their categories F(Δ) are essentially the same. Then,using this reduction, B. Deng proved that F(RΔ) is finite if all the indecomposable modules inF(RΔ) are RΔ-directing.

The generalization of those results, obtained by B. Deng, to standardly stratified algebrasare not so obviously. First, B. Deng used the fact that the Cartan matrix of a quasi-hereditaryalgebra R is invertible, and it is the case since R has finite global dimension; however, in gen-eral, the global dimension of a standardly stratified algebra is infinity. Second, B. Deng uses the“Process of standardization” given by V. Dlab and C.M. Ringel in [7] which is only valid for

752 E.D.N. Marcos et al. / Journal of Algebra 302 (2006) 750–770

quasi-hereditary algebras; in the case of standardly stratified algebras, we have to use a moregeneralized “process of standardization.”

So, we have to look for a more generalized “process of standardization” for standardly strat-ified algebras. Fortunately, that process exists, and that is why the stratifying systems appears inthis work. On the one hand, the stratifying systems generalizes the “Process of standardization”given by V. Dlab and C.M. Ringel in [7] (see Theorem 3.2 in [14]). On the other hand, stratifyingsystems generalizes the standard modules RΔ. Then, we go further and consider the problemdirectly for stratifying systems. One of the main result in this paper is the following: “if (θ,�) isa stratifying system and F(θ) is θ -directing, then F(θ) is of finite representation type.”

In the paper, we consider the quadratic form qθ associated to a stratifying system (θ,�). Later,we study, for standardly stratified algebras, RΔ-directing and RΔ-omnipresent modules in thecategory F(RΔ), and show that the existence of one of such modules X implies that all modulesin F(RΔ) have projective dimension at most 2, and also that X has projective (respectivelyrelative injective) dimension at most 1. On the other hand, using the Ext-projective stratifyingsystems (θ,Q,�) introduced in [14], we show that the study of θ -directing modules in F(θ) canbe reduced to the study of those modules over certain standardly stratified algebra B that admitsa BΔ-directing and BΔ-omnipresent module. Moreover, we also prove that the quadratic formsqθ and qBΔ are essentially the same. Finally, using this reduction, we prove that indF(θ) is finiteif all the indecomposable modules in F(θ) are θ -directing.

1. Preliminaries

Throughout the paper, we denote by [1, t] the set {1,2, . . . , t} and by � a total order on [1, t].However, we reserve the notation � for the natural order on [1, t].

Let R be an algebra. We start this section by recalling the definition of stratifying system, Ext-injective stratifying system and Ext-projective stratifying system given in [13,14]. Afterwards,we recall the notion of standard stratifying system. Finally, we introduce briefly the notion ofrelative projective dimension, X -resolution dimension and Euler’s quadratic form.

Definition 1.1. [13] A stratifying system (θ,�) of size t consists of a set θ = {θ(i)}ti=1 of inde-composable R-modules and a total order � on [1, t], satisfying the following conditions:

(a) HomR(θ(j), θ(i)) = 0 for j � i,(b) Ext1R(θ(j), θ(i)) = 0 for j � i.

In the theory of stratifying systems, there are three equivalent notions: (a) Stratifying systems(see 1.1), (b) Ext-injective stratifying system (see 1.2), and (c) Ext-projective stratifying system(see 1.3). The equivalence of those notions implies in particular that, given a stratifying system(θ,�) of size t , we can associate to it an uniquely determined Ext-injective stratifying system(eiss for short) (θ,Y ,�) and an uniquely determined Ext-projective stratifying system (epss forshort) (θ,Q,�). Moreover, the set Y = {Y(1), . . . , Y (t)} (respectively Q = {Q(1), . . . ,Q(t)})consists of pairwise non-isomorphic indecomposable R-modules. In order to simplify the state-ments, we set Y = ∐ti=1 Y(i) and Q = ∐ti=1 Q(i). Finally, we introduce the categories P(θ)and I(θ), which will be used throughout the paper


P(θ) = {X ∈ mod R: Ext1R(X.−)∣∣F(θ) = 0} andI(θ) = {X ∈ mod R: Ext1R(−,X)∣∣F(θ) = 0}.

Definition 1.2. [8] Let θ = {θ(i)}ti=1 be a set of non-zero R-modules, Y = {Yi}ti=1 a set ofindecomposable R-modules, and � a total order on [1, t]. The triple (θ,Y ,�) is an Ext-injectivestratifying system of size t if the following three conditions hold:

(a) HomR(θ(j), θ(i)) = 0 for j � i,(b) for each i ∈ [1, t], there is an exact sequence 0 → θ(i) αi−→ Y(i) → Z(i) → 0 such that

Z(i) ∈F({θ(j): j ≺ i}),(c) Ext1R(−, Y )|F(θ) = 0.

Definition 1.3. [14] Let θ = {θ(i)}ti=1 be a set of non-zero R-modules, Q = {Q(i)}ti=1 a set ofindecomposable R-modules and � a total order on [1, t]. The triple (θ,Q,�) is an Ext-projectivestratifying system of size t if the following three conditions hold:

(a) HomR(θ(j), θ(i)) = 0 for j � i,(b) for each i ∈ [1, t], there is an exact sequence 0 → K(i) → Q(i) βi−→ θ(i) → 0 such that

K(i) ∈ F({θ(j): j � i}),(c) Ext1R(Q,−)|F(θ) = 0.

Let (θ,�) be a stratifying system. K. Erdmann and C. Saenz have shown in [8] that thefiltration multiplicities [M : θ(i)] do not depend on the filtration of M ∈F(θ). For this reason, wecan introduce the θ -support of M as the set Suppθ (M) = {i ∈ [1, t]: [M : θ(i)] = 0}. Therefore,Suppθ (M) is empty if M = 0. We define the functions min,max :F(θ) → [1, t] ∪ {±∞} asfollows: (a) min(0) := +∞ and max(0) := −∞, and (b) min(M) := min(Suppθ (M),�) andmax(M) := max(Suppθ (M),�) if M = 0. Finally, we recall that a stratifying system (θ,�) ofsize t is standard if RR ∈ F(θ).

Let X be a class of R-modules. We denote by X∧ the subcategory of mod R whose objectsare those R-modules X for which there exists a finite X -resolution. That is, M ∈ X∧ if andonly if there exists a long exact sequence 0 → Xn → ·· · → X1 → X0 → M → 0 with Xi ∈ Xfor all i = 0,1, . . . , n. Dually, X∨ is the subcategory of mod R whose objects have a finiteX -coresolution. We denote by pdX the projective dimension of X. Similarly, we use the notationidX for the injective dimension of X. Following Auslander and Buchweitz in [1], we recall theconcepts of relative projective dimension and relative resolution dimension of a given module.

Definition 1.4. Let X be a class of objects in mod R, and M be an R-module.

(a) We shall denote by pdX M the relative projective dimension of M with respect to X . Thatis, pdX M := −∞ if M = 0, and pdX M := min{n: ExtjR(M,−)|X = 0 for any j > n} ifM = 0. Dually, idX M is the relative injective dimension of M with respect to X .

(b) We shall denote by resdimX M the X -resolution dimension of M . That is, resdimX M :=−∞ if M = 0, resdimX M := +∞ if M /∈ X∧, and resdimX M := min{r: there is an exactsequence 0 → Xr → ·· · → X0 → M → 0, with Xi ∈ X for any i} if M ∈ X∧. Dually, wedenote by coresdimX M the X -coresolution dimension of M .


(c) For any class C of R-modules, we set pdX C := sup{pdX M: M ∈ C} and resdimX C :=sup{resdimX M: M ∈ C}.

Let A be an algebra, and s be the number of iso-classes of simple A-modules. Let F(A)be the free abelian group with basis the set of isomorphism classes in mod A, and R(A) bethe subgroup generated by the formal sums M ′ − M + M ′′ for any exact sequence 0 → M ′ →M → M ′′ → 0 in mod A. By definition, the Grothendieck group K0(A) of A is the quotientF(A)/R(A). By the Jordan–Holder’s theorem, it is known that K0(R) is a free abelian groupwith basis e(i) = dimS(i), where S(1), S(2), . . . , S(s) are the iso-classes of simple A-modules.Using this basis, we may identify K0(A) with Zs . Then, we can associate to each A-moduleM the dimension vector dimM = ∑si=1[M : S(i)]e(i) ∈ Zs . We will denote by CA the Cartanmatrix of A, which is a s × s matrix with ij entry equals to dimk HomA(P (i),P (j)). Thus, thej th column is given by dimP(j)T . If CA is invertible, then C

−TA defines a bilinear form 〈−,−〉A

on K0(A,Q) := K0(A)⊗Z Q given by 〈x, y〉A := xC−TA yT . This bilinear form has the followinghomological interpretation.

Lemma 1.5. [12] Assume that the Cartan matrix CA is invertible. If pdM < ∞ or idN < ∞then

〈dimM,dimN〉A =∑t�0

(−1)t dimk ExttA(M,N).

If the Cartan matrix of A is invertible, then we have the Euler’s quadratic form χA(x) :=

〈x, x〉A on K0(A,Q). Hence, if pdM < ∞ or idM < ∞ then

χA(dimM) =

∑t�0

(−1)t dimk ExttA(M,M).

2. Quadratic forms on K0(F(θ))

Let R be an algebra, and (θ,�) be a stratifying system of size t . Let F(F(θ)) be the freeabelian group with basis the set of isomorphism classes of objects in F(θ), and R(F(θ)) be thesubgroup generated by the formal sums M ′ −M +M ′′ for any exact sequence 0 → M ′ → M →M ′′ → 0 in F(θ). We have by definition that the Grothendieck group K0(F(θ)) of F(θ) is thequotient F(F(θ))/R(F(θ)).

Lemma 2.1. K0(F(θ)) is a free abelian group of rank t with basis the images [θ(i)] of θ(i)under the canonical map F(F(θ)) → K0(F(θ)).

Proof. It follows from the fact that for any M ∈ F(θ) the filtration multiplicities [M : θ(i)] donot depend on a given filtration of M in θ (see [8]). �Remark 2.2. As we have seen above, the relative simple modules θ in F(θ) generate the freegroup K0(F(θ)) because of the fact that the filtration multiplicities [M : θ(i)] do not depend ona given filtration of M in θ . Moreover, we know that F(θ) is a functorially finite subcategory ofmod R. However, it is not true that for any functorially finite subcategory C of mod R we have


that the relative simple modules have a “good behavior” as above. Indeed, let R be the hereditarypath algebra given by the following quiver

1

ρ

θ3

α β

2ξ

4

.

Consider the R-modules: M := P(1)/〈ξρ − αθ〉, K1 := P(1)/〈ξρ,αθ〉 and K2 := P(1)/〈ξρ − αθ, ξρ − βθ〉. Let C := add{S(4),M,K1,K2}. We have that the relative simple modulesin C are S(4), K1 and K2 since the only proper submodule of K1, K2 and M lying in C is S(4)and the quotients K1/S(4), K2/S(4) do not belong to C. On the other hand, we have two exactsequences 0 → S(4) → M → Ki → 0 for i = 1,2. Since K1 � K2, we get that M admits twodifferent filtrations; and so there is no unicity of filtrations of M with the relative simple modulesin C.

Using the basis dim θ θ(i) := [θ(i)], we may identify K0(F(θ)) with Zt . Hence, for anyM ∈ F(θ), we have the dimension vector

dim θ M :=t∑

i=1

[M : θ(i)]dim θ θ(i) ∈ Zt .

Definition 2.3. Associated to a stratifying system (θ,�) of size t , we have two quadratic forms:

(a) the Tits form qθ : Zt → Z defined by the equality

qθ (x) :=2∑

l=0

∑i,j

(−1)l dimk ExtlR(θ(i), θ(j)

)x(i)x(j),

(b) if pdF(θ) θ < ∞, we have a bilinear form 〈−,−〉θ on Zt defined by

〈x, y〉θ :=∑l�0

∑i,j

(−1)l dimk ExtlR(θ(i), θ(j)

)x(i)y(j).

In this case, the Euler quadratic form χθ : Zt → Z is by definition

χθ (x) := 〈x, x〉θ .

In 2.3(b), we needed pdF(θ) θ to be finite. So, it would be useful to have a condition on θ toget a bound for pdF(θ) θ . The following theorem establishes a bound for this number in termsof the category I(θ). We recall that, a given subcategory X of mod R is called coresolving if Xsatisfies the following three conditions: (a) closed under extensions, (b) closed under cokernelsof injections and (c) contains the injective R-modules. Observe that conditions (a) and (c) arevalid for X = I(θ).


Theorem 2.4. [16] Let (θ,�) be a stratifying system of size t , and s be the number of iso-classesof simple R-modules. If I(θ) is coresolving then

pdF(θ) � t � s and pdF(θ)F(θ) � t − 1.

Let (θ,�) be a stratifying system of R-modules of size t , and (θ,Q,�) be the epss associ-ated to (θ,�). We introduce the t × t matrices D, CQ and DQ as follows: (a) D = (dij ), wheredij := dimk HomR(Q(i), θ(j)), (b) the ij entry of CQ is equal to dimk HomR(Q(i),Q(j)),and (c) the j th column of DQ is given by dim θ Q(j)

T . We recall that the functor eQ :=HomR(Q,−) :F(θ) → F(BΔ) is an equivalence of exact categories, where B := End(RQ)opis a standardly stratified algebra and {ε1, ε2, . . . , εt } is a fixed complete set of primitive orthogo-nal idempotents of B such that Bεi � eQ(Q(i)) for all i (see Theorem 3.2 in [14]).

Lemma 2.5. With the notation introduced above we have:

(a) the matrix CQ is equal to the Cartan matrix CB of B ,(b) the matrix D is upper triangular and detD = ∏ti=1 dimk End(Rθ(i)),(c) the matrix DQ is lower triangular and detDQ = 1,(d) CQ = DDQ,(e) dim eQ(M) = dim θ MDT for any M ∈F(θ).

Proof. The item (a) follows directly from the equivalence eQ :F(θ) →F(BΔ). Let Di be i-rowof the matrix D. By the equality (see Lemma 2.6 in [14])

dimk HomR(Q(i),M

) =t∑

j=1

[M : θ(j)]dimk HomR(Q(i), θ(j)),

we have dimk HomR(Q(i),M) = ∑tj=1[M : θ(j)]dij = Di(dim θ M)T . Hence (d) and (e) fol-low.

On the other hand, DQ(ij) = [Q(j) : θ(i)] = 0 for i < j . So DQ is lower triangular anddetDQ = ∏ti=1[Q(i) : θ(i)] = 1, proving (c). To prove (b) we have

dij = dimk HomR(Q(i), θ(j)

) = dimk HomB(BP (i), BΔ(j)) = [BΔ(j), S(i)].Thus dij = 0 for i > j , and so D is upper triangular. Finally, by Lemma 2.6 in [14], we have thatdii = dimk End(Rθ(i)). Hence, detD = ∏ti=1 dimk End(Rθ(i)). �Definition 2.6. Associated to an epss (θ,Q,�) of size t , we have a bilinear form 〈−,−〉Q on Zt ,where 〈x, y〉Q := x(D−TQ D)yT .

Proposition 2.7. For any M,N ∈F(θ), we have that

(a) 〈dim θ M,dim θ N〉Q = 〈dim eQ(M),dim eQ(N)〉B ,(b) 〈dim θ M,dim θ N〉Q = 〈dimBΔ eQ(M),dimBΔ eQ(N)〉BΔ.


Proof. (a) By (a) and (d) in 2.5, we get 〈x, y〉B = x(D−T D−TQ )yT . Hence, the result followsfrom 2.5(e).

(b) From (a), we have 〈dim θ θ(i),dim θ θ(j)〉Q = 〈dim BΔ(i),dim BΔ(j)〉B . Then,〈dim θ θ(i),dim θ θ(j)〉Q =∑

l�0(−1)l dimk ExtlB(BΔ(i), BΔ(j)) (see 1.5), proving the re-sult. �Theorem 2.8. If R is a standardly stratified algebra, then the Cartan matrix CR is invertible.Moreover, the following equality holds for any M,N ∈ F(RΔ)

〈dimRΔ

M,dimRΔ

N〉RΔ = 〈dimM,dimN〉R.

Proof. Let (RΔ,Q,�) be the epss associated to (RΔ,�). Since the canonical stratifying system(RΔ,�) is standard, we have that Q = RR. Then B := End(RQ)op = R; therefore, by 2.5, weget detCR = ∏si=1 dimk End(RΔ(i)) = 0, proving that CR is invertible. Then, from the previousproposition, we get the result. �3. General facts about θ -directing modules

Let R be an algebra. We recall, from C.M. Ringel in [12], the following definitions. Let Cbe a full subcategory of mod R which is closed under direct summands. A path in C is a finitesequence (X0,X1, . . . ,Xm) of indecomposable modules in C such that rad(Xi−1,Xi) = 0 forall 1 � i � m, where rad(Xi−1,Xi) is the set of all non-invertible morphisms from Xi−1 to Xi .We write M �C N to indicate that there is a path from M to N in C. If m � 1 and X0 � Xm,then the path (X0,X1, . . . ,Xm) is called a cycle in C. An indecomposable module X in C iscalled C-directing if X does not occur in a cycle in C. Furthermore, the category C is said to beC-directing if any X ∈ indC is C-directing. Finally, we say that C is directing if X is mod R-directing for any X ∈ indC. Observe that, if C is directing then C is C-directing, but the converseis false.

Let (θ,�) be a stratifying system of size t . Since the category F(θ) is closed under extensionsand direct summands (see [13]), we can set C = F(θ) in the definitions above. To make simple,we replace the expression “F(θ)” by “θ .” Thus, we get the definition of M �θ N and θ -directing.On the other hand, we say that M ∈F(θ) is θ -omnipresent if [M : θ(i)] = 0 for any i.

Definition 3.1. For any M ∈ indF(θ), we set (�θ ,M) := {X ∈ indF(θ): X �θ M} and(�θ ,M] := (�θ ,M) ∪ {M}. Similarly, we define (M,�θ ) := {X ∈ indF(θ): X �θ M} and[M,�θ ) := (M,�θ ) ∪ {M}.

Lemma 3.2. Let X,N ∈ indF(θ). If Ext1R(X,N) = 0 then N �θ X.

Proof. Assume that Ext1R(X,N) = 0. Consider a non-split exact sequence

0 → N g−→ E f−→ X → 0. (1)

Let E′ be an indecomposable direct summand of E. We denote by π :E → E′ and ı :E′ → E,respectively, the canonical projection and inclusion. Since the sequence (1) is non-split, we havethat g ∈ rad(N,E) and f ∈ rad(E,X). Hence πg ∈ rad(N,E′) and f ı ∈ rad(E′,X).


We assert that πg = 0. Indeed, suppose that πg = 0, so there exists ϕ :X → E′ such thatϕf = π and ϕf ı = πı = 1E′ . Then ϕ is a split epimorphism, and therefore it has to be an isomor-phism since X is indecomposable. Thus, (1) splits giving a contradiction, proving that πg = 0.Likewise, it can be proven that f ı = 0. Then we get that N �θ X since F(θ) is closed underdirect summands and extensions. �Corollary 3.3. If X is θ -directing then Ext1R(X,X) = 0.

Proof. It follows from 3.2. �In the general situation of mod R, it is well known that if [M : S(i)] = 0 for some i, then

HomR(P (i),M) = 0 and HomR(M, I (i)) = 0, where P(i) is the projective cover and I (i) isthe injective envelope of the simple module S(i). The following lemma will be used to prove ageneralization of that fact for the category F(θ) (see 3.5). In this case, θ(i) is a simple object ofF(θ), Q(i) the “Ext-projective” cover and Y(i) the “Ext-injective” envelope of θ(i) in F(θ).

Lemma 3.4. If M ∈ F(θ) then[M : θ(i)]dimk EndR(θ(i)) � min{dimk HomR(Q(i),M),dimk HomR(M,Y(i))}.

Proof. The result follows from the following equalities (see Lemma 2.6 in [14])

dimk HomR(Q(i), θ(i)

) = dimk EndR(θ(i)) = dimk HomR(θ(i), Y (i)),dimk HomR

(Q(i),M

) =t∑

j=1

[M : θ(j)]dimk HomR(Q(i), θ(j)),

dimk HomR(M,Y(i)

) =t∑

j=1

[M : θ(j)]dimk HomR(θ(j), Y (i)). �

Corollary 3.5. If [M : θ(i)] = 0 then HomR(Q(i),M) and HomR(M,Y (i)) are non-zero.

Proof. It follows from 3.4. �Proposition 3.6. Let X ∈F(θ). Then

(a) there is an exact sequence 0 → X′ → Q0 εX−→ X → 0 in F(θ) such that Q0 ∈ addQ, andεX is the right minimal P(θ)-approximation of X,

(b) if X is indecomposable and X /∈ addQ then Ker εX ∈ add(�θ ,X),(c) if N ∈ indF(θ) is θ -omnipresent then Ker εX ∈ add(�θ ,N).

Proof. If Q(j) is a direct summand of Q0, we will denote by πj :Q0 → Q(j) the canonicalprojection. Let Z be an indecomposable direct summand of Ker εX . Since Z ⊆ Q0 ∈ addQ, wehave some direct summand Q(j) of Q0 such that πj (Z) = 0. Let gZ := πj |Z :Z → Q(j). Then,we have that 0 = gZ ∈ rad(Z,Q(j)) since εX :Q0 → X is right-minimal.

(a) It is Proposition 2.10 in [14].


(b) Assume that X is indecomposable, and X /∈ addQ. Let Z be an indecomposable directsummand of Ker εX . Hence, as we have seen above, we have a morphism gZ :Z → Q(j) suchthat 0 = gZ ∈ rad(Z,Q(j)). Let ε′ := εX|Q(j), so we have that 0 = ε′ ∈ rad(Q(j),X) sinceε :Q0 → X is right minimal and X /∈ addQ. Therefore, we have the path Z gZ−→ Q(j) ε′−→ X inF(θ), proving that Ker εX ∈ add(�θ ,X).

(c) Let N ∈ indF(θ) be θ -omnipresent, and let Z be an indecomposable direct summand ofKer εX . Then we have a morphism gZ :Z → Q(j) such that 0 = gZ ∈ rad(Z,Q(j)). Using thatN is θ -omnipresent, we get from 3.5 that HomR(Q(j),N) = 0 for any j . Let f :Q(j) → N bea non-zero morphism. Thus, we get that Z �θ N since 0 = g ∈ rad(Z,Q(j)) and f = 0, provingthat Ker εX ∈ add(�θ ,N). �

The following result will be used in 3.12 to prove that, under certain conditions on θ , thevector dim θ N is a positive root of qθ .

Lemma 3.7. Let N ∈ indF(θ) be θ -directing. Then

(a) End(RN) � k,(b) if idaddQ(N) = 0 then id(�θ ,N ](N) = 0,(c) if pdaddY (N) = 0 then pd[N,�θ )(N) = 0.

Proof. (a) Since N is indecomposable and the field k is algebraically closed, we get thatEnd(RN)/ rad End(RN) � k. Furthermore, using that N is θ -directing, we conclude thatrad End(RN) = 0, proving that End(RN) � k.

(b) Let idaddQ(N) = 0. We prove by induction on j that ExtjR(−,N)|(�θ ,N ] = 0 for all j � 1.Suppose there is X ∈ (�θ ,N] such that Ext1R(X,N) = 0. Then by 3.2, we obtain that N �θ X,

and so N has to be θ -directing which is a contradiction. Hence Ext1R(−,N)|(�θ ,N ] = 0. Assumeby induction that ExtjR(−,N)|(�θ ,N ] = 0 for j � 2. Then we prove that Extj+1R (−,N)|(�θ ,N ] = 0.Let X ∈ (�θ ,N ], so we may assume that X /∈ addQ since idaddQ(N) = 0. By 3.6, there existsan exact sequence

0 → X′ → Q0 → X → 0 with Q0 ∈ addQ and X′ ∈ add(�θ ,N ]. (2)

Thus ExtjR(X′,N) = 0 for j � 2. Applying the functor HomR(−,N) to the exact sequence in (2),

we get the following exact sequence

ExtjR(Q0,N) → ExtjR(X′,N) → Extj+1R (X,N) → Extj+1R (Q0,N).

Therefore, ExtjR(X′,N) ∼→ Extj+1R (X,N) for j � 1 since idaddQ(N) = 0. Then,

Extj+1R (X,N) = 0 because of the equality ExtjR(X′,N) = 0. Consequently, we obtain the equal-ity Extj+1R (−,N)|(�θ ,N ] = 0, proving that id(�θ ,N ](N) = 0.

(c) It is similar to (b). �Let (θ,�) be a stratifying system, and (θ,Q,�) be the epss associated to (θ,�) (see 1.3). We

recall from [14] that F(θ) ∩P(θ) = addQ, where P(θ) = {X ∈ mod R: Ext1R(X,−)|F(θ) = 0}.Moreover, due to Corollary 1.11 in [13], we have that the category F(θ) is closed under direct


summands and is functorially finite. So, by [11], we get that F(θ) has relative Auslander–Reitensequences. Furthermore, if X belongs to indF(θ) and X /∈ addQ, then we denote by τθX theleft-hand term in the relative Auslander–Reiten sequence 0 → τθX → E → X → 0 in F(θ). Onthe other hand, if (θ,Y ,�) is the eiss (see 1.2) associated to (θ,�), we recall that F(θ)∩I(θ) =addY , where I(θ) = {X ∈ mod R: Ext1R(−,X)|F(θ) = 0} (see [13]).

The following series of lemmas and propositions are given in order to prove 3.12, which isthe main result in this section.

Lemma 3.8. Let N ∈ indF(θ) be θ -directing and θ -omnipresent. Then, for any M ∈ (�θ ,N],we have the following isomorphism

Ext1R(M,−)|F(θ) ∼→ D HomR(−, τθM)|F(θ).

Proof. Let M ∈ (�θ ,N]. We may assume that M /∈ addQ since τθQ(j) = 0 for any j , andExt1R(Q,−)|F(θ) = 0. Using that F(θ) has relative Auslander–Reiten sequences (see [11]), weobtain from Corollary 9.4 in [10] that

Ext1R(M,Z)∼→ D(HomR(Z, τθM)/Iθ (Z, τθM)) for any Z ∈ F(θ),

where Iθ (Z, τθM) is the set of morphisms f :Z → τθM which factor through some objectof addY . Using that N is θ -directing and θ -omnipresent, it can be seen, by using 3.5, thatHomR(Y, τθM) = 0. Thus Iθ (Z, τθM) = 0, proving the result. �Proposition 3.9. Let N ∈ indF(θ) be θ -directing and θ -omnipresent.

If Ext2R(Q,Y ) = 0 then

(a) Ext2R(M,−)|F(θ) = 0 for any M ∈ (�θ ,N],(b) resdimaddQ(�θ ,N] � 1.

Proof. (a) Assume that Ext2R(Q,Y ) = 0. By 3.8, we have that Ext1R(M,−) is right exact onF(θ) since HomR(−, τθM) is a left exact functor. We assert that

Ext2R(−, Y )|F(θ) = 0. (3)

Indeed, let M ∈ F(θ). Then by 3.6(a), we have an exact sequence 0 → K → Q0 → M → 0 inF(θ) with Q0 ∈ addQ. Applying the functor HomR(−, Y ) to this sequence, we get the exactsequence Ext1R(K,Y ) → Ext2R(M,Y ) → Ext2R(Q0, Y ). Then we get that Ext2R(M,Y ) = 0 sinceExt1R(K,Y ) = 0 = Ext2R(Q0, Y ), proving that Ext2R(−, Y )|F(θ) = 0.

Suppose that M ∈ (�θ ,N] and X ∈ F(θ). Then by Lemma 1.5 in [8], we get an exact se-quence 0 → X → Y0 → Z → 0 in F(θ) with Y0 ∈ addY . Applying the functor HomR(M,−)to this sequence and using that Ext1R(M,−) is right exact on F(θ), we get the following exactsequence

0 → Ext2R(M,X) → Ext2R(M,Y0).

Thus, by (3), we have that Ext2 (M,X) = 0, proving that Ext2 (M,−)|F(θ) = 0.

R R


(b) Let M ∈ (�θ ,N]. By 3.6(a), we have an exact sequence 0 → M ′ → Q0 → M → 0 withM ′ ∈ F(θ) and Q0 ∈ addQ. For any Z ∈ F(θ), we apply the functor HomR(−,Z) to this exactsequence to get the exact sequence

Ext1R(Q0,Z) → Ext1R(M ′,Z) → Ext2R(M,Z).

Since Ext1R(Q0,Z) = 0 = Ext2R(M,Z) for any Z ∈ F(θ), we obtain that Ext1R(M ′,−)|F(θ) = 0.Thus M ′ ∈ F(θ) ∩P(θ) = addQ, and so resdimaddQ M � 1. �Lemma 3.10. Let (θ,�) be a stratifying system. Then

(a) pdF(θ) F(θ) = pdF(θ) θ and pdF(θ) = pd θ ,(b) if idaddQ(F(θ)) = 0 then pdF(θ)(M) = resdimaddQ(M) for any M ∈F(θ).

Proof. (a) We only prove the first part of the statement, for the second one is quite similar. Let(θ,�) be a stratifying system of size t . We may assume that pdF(θ) θ = m < ∞. Let 0 = X ∈F(θ) and i := minX. We prove, by reverse induction on i, that pdF(θ) X � m.

If maxX = i then X � θ(i)mi , and so pdF(θ) X � m. Assume that maxX � i. Then, byProposition 2.9 in [14], we have an exact sequence

0 → X′ → X → θ(i)mi → 0 with minX′ � i. (4)

For M ∈F(θ), we apply the functor HomR(−,M) to the exact sequence in (4) to get the follow-ing exact sequence:

ExtjR(θ(i)mi ,M

) → ExtjR(X,M) → ExtjR(X′,M) → Extj+1R (θ(i)mi ,M).Using that pdF(θ) θ = m, we get ExtjR(X,M)

∼→ ExtjR(X′,M) for all j � m + 1. Hence, byreverse induction and the fact that minX′ � i, we get the equality ExtjR(X′,M) = 0 for all j �m + 1. Then ExtjR(X,M) = 0 for all j � m + 1, proving that pdF(θ) X � m.

(b) Assume that idaddQ(F(θ)) = 0. Since pdF(θ)(addQ) = 0 and addQ ⊆ F(θ), we ob-tain, by the dual of Theorem 2.1 in [16], the equality pdF(θ)(M) = resdimaddQ(M) for anyM ∈ (addQ)∧. Then, the result follows since, by Corollary 2.11 in [14], we know that F(θ) ⊆(addQ)∧. �

In [12], C.M. Ringel proved (see 2.4 (7)) that if N is a sincere and directing R-module, thenpdN � 1, idN � 1 and gl.dimR � 2. In the following proposition, we generalize to “relativetheory” in F(θ) this result.

Proposition 3.11. Let N ∈ indF(θ) be θ -directing and θ -omnipresent.

(a) If idaddQ(F(θ)) = 0, then we have the following inequalities

idF(θ) N � 1, pdF(θ)(�θ ,N] � 1 and pdF(θ) F(θ) � 2.


(b) If pdaddY (F(θ)) = 0, then we have the following inequalitiespdF(θ) N � 1, idF(θ)[N,�θ ) � 1 and pdF(θ) F(θ) � 2.

Proof. We only prove (a) since the proof of (b) can be obtained from (a) by duality.Assume that idaddQ(F(θ)) = 0. We start by proving that ExtjR(−,N)|F(θ) = 0 for all j � 2.

Indeed, let X ∈ F(θ), and 0 → X′ → Q0 → X → 0 be the exact sequence of 3.6(a). Hence,by 3.6(c), we have that X′ ∈ add(�θ ,N]. Then, from 3.7(b), we get ExtjR(X′,N) = 0 for anyj � 1. Applying the functor HomR(−,N) to the above sequence, we obtain the following exactsequence

ExtjR(Q0,N) → ExtjR(X′,N) → Extj+1R (X,N) → Extj+1R (Q0,N).

Therefore, ExtjR(X′,N) ∼→ Extj+1R (X,N) for any j � 1 since idaddQ (F(θ)) = 0. Thus,

Extj+1R (−,N)|F(θ) = 0 for all j � 1, proving that idF(θ) N � 1. The next aim is to prove theinequality pdF(θ)(�θ ,N] � 1; however, it follows easily from 3.9(b) and 3.10(b).

Finally, we prove that ExtjR(θ(i),−)|F(θ) = 0 for all j � 3, which is enough to get the resultin view of 3.10(a). To do that, we fix some i and consider the canonical exact sequence (see 1.3)0 → K(i) → Q(i) → θ(i) → 0. Thus, since N is θ -omnipresent, we obtain from 3.6(c) thatK(i) ∈ add(�θ ,N ]. Due to pdF(θ)(�θ ,N] � 1, we get that ExtjR(K(i),−)|F(θ) = 0 for anyj � 2.

Applying the functor HomR(−,N) to the canonical exact sequence, for any Z ∈F(θ), we getthe exact sequence

ExtjR(Q(i),Z

) → ExtjR(K(i),Z) → Extj+1R (θ(i),Z) → Extj+1R (Q(i),Z).Thus ExtjR(K(i),Z)

∼→ Extj+1R (θ(i),Z) for all j � 1. Hence Extj+1R (θ(i),−)|F(θ) = 0 for allj � 2, proving that pdF(θ) θ � 2. �

The following is the main result in this section. As an application, we get 3.13, which playsan important role in the proof of the main results in Section 4.

Theorem 3.12. Let (θ,�) be a stratifying system such that pdQ � 1 or I(θ) is a coresolvingsubcategory of mod R. If there is some X ∈ indF(θ) that is θ -directing and θ -omnipresent, then

(a) max(pdF(θ) X, idF(θ) X) � 1 and pdF(θ) F(θ) � 2,(b) if χθ (dim θ X) = χR(dimX) then qθ (dim θ X) = 1.

Proof. Assume that pdQ � 1 or I(θ) is coresolving.(a) If pdQ � 1, then the hypothesis needed in 3.11(a) holds since we know that

Ext1R(Q,−)|F(θ) = 0, and so (a) follows. Otherwise, if I(θ) is coresolving, then we have byProposition 3.8(b) in [15] that ExtjR(F(θ),I(θ)) = 0 for any j > 0. Hence, the hypothesisneeded in 3.11(b) holds, proving (a).

(b) Suppose that χθ (dim θ X) = χR(dimX). Then, by (a), we have that qθ (dim θ X) =χθ (dim θ X). On the other hand, from 3.7, we get χR(dimX) = 1, proving that qθ (dim θ X) = 1.�


Corollary 3.13. Let R be a standardly stratified algebra. If there is an X ∈ indF(RΔ) that isRΔ-directing and RΔ-omnipresent, then

max(pdX, idF(RΔ) X) � 1, pdF(RΔ) � 2 and qRΔ(dimRΔ X) = 1.

Proof. Let (RΔ,Q,�) be the epss associated to (RΔ,�). Since R is a standardly stratifiedalgebra, we have that Q = RR. Moreover, due to the fact that pdF(RΔ) is finite, we get from3.10(b) that pdF(RΔ) M = pdM for any M ∈ F(RΔ). On the other hand, by 2.8, we know thatχ

RΔ(dimRΔ X) = χR(dimX). Then, the result is a consequence of 3.12. �4. Main results

We start this section by fixing some notation that will be used throughout. Let (θ,�) be astratifying system. In the following, we make a construction, given in [14], to reduce each mod-ule X ∈ F(θ) to a BXΔ-omnipresent module over a standardly stratified algebra BX . Indeed, forany 0 = X ∈ F(θ), we set θX := {θ(i): i ∈ Suppθ (X)}, where Suppθ (X) := {i: [X : θ(i)] = 0}.Then, we have the induced stratifying system (θX,�). Consequently, we have that F(θX) ⊆F(θ), and that X is θX-omnipresent. We denote by (θX,QX,�) and (θX,YX,�) the epss andthe eiss associated, respectively, to (θX,�). Also, we have that F(θX) ∩ P(θX) = addQXand F(θX) ∩ I(θX) = addYX . Furthermore, we have that the endomorphism algebra BX =End(RQX)op is standardly stratified, and the functor eQX = HomR(QX,−) :F(θX) → F(BXΔ)is an equivalence of exact categories (see Theorem 3.2 in [14]). On the other hand, we have thatX′ := eQX(X) is BXΔ-omnipresent. Moreover, if X is θ -directing then X′ is also BXΔ-directing.

Lemma 4.1. Let X ∈ indF(θ) be θ -directing and such that Ext2R(Q,YX) = 0. Then, for anyi, j ∈ Suppθ (X) and = 0,1,2, we have the following equality

dimk Ext

R

(θ(i), θ(j)

) = dimk ExtBX(BXΔ(i), BXΔ(j)

).

Proof. Let i, j ∈ Suppθ (X). Since the functor eQX :F(θX) → F(BXΔ) is an equivalence of ex-act categories, we obtain the equality dimk ExtR(θ(i), θ(j)) = dimk ExtBX(BXΔ(i), BXΔ(j)) for

= 0,1. In order to prove that the equality holds for = 2, we use the canonical exact sequence0 → KX(i) → QX(i) → θX(i) → 0 given in 1.3. Applying the functor HomR(−, θX(j)) to thatsequence, and since Ext1R(QX(i), θX(j)) = 0, we get the following exact sequence

0 → Ext1R(KX(i), θX(j)

) → Ext2R(θX(i), θX(j)) → Ext2R(QX(i), θX(j)). (5)We assert that Ext2R(QX(i), θX(j)) = 0. Indeed, suppose that Ext2R(QX(i), θX(j)) = 0. Apply-ing the functor HomR(QX(i),−) to the exact sequence 0 → θX(j) → YX(j) → ZX(j) → 0that is given in 1.2, we get the following exact sequence 0 → Ext2R(QX(i), θX(j)) →Ext2R(QX(i), YX(j)). Hence Ext

2R(QX(i), YX(j)) = 0, and so QX /∈ addQ since

Ext2R(Q,YX) = 0. Due to QX(i) ∈F(θ), we get from 3.6(a) the following exact sequence

0 → K → Q0 ϕ−→ QX(i) → 0 with Q0 ∈ addQ, K ∈F(θ), (6)and ϕ :Q0 → QX(i) is the right minimal P(θ)-approximation of QX(i). Applying the func-tor HomR(−, YX(j)) to (6) and using that Ext1 (Q0, YX(j)) = 0 and Ext2 (Q0, YX(j)) = 0,

R R


we get Ext1R(K,YX(j))∼→ Ext2R(QX(i), YX(j)). Then, Ext1R(K,YX(j)) = 0 since

Ext2R(QX(i), YX(j)) = 0, and so there is an indecomposable direct summand K ′ of K withExt1R(K

′, YX(j)) = 0. Hence, by 3.2 and 3.6(b), we obtain that YX(j) �θ K ′ �θ QX(i). Onthe other hand, since X is θX-omnipresent, we get from 3.5 that HomR(X,YX(j)) = 0 andHomR(QX(i),X) = 0. Then X belongs to a cycle in F(θ), contradicting that X is θ -directing.So we have that Ext2R(QX(i), θX(j)) = 0, and then by (5), we obtain the following isomorphism

Ext1R(KX(i), θX(j)

) � Ext2R(θX(i), θX(j)). (7)Finally, applying the functor HomBX(−, BXΔ(j)) to the exact sequence

0 → eQX(KX(i)

) → eQX(QX(i)) → BXΔ(i) → 0,we have the following isomorphism

Ext2BX(BXΔ(i), BXΔ(j)

) � Ext1BX(eQX

(KX(i)

), BXΔ(j)

). (8)

Then the result follows by using (7), (8) and the fact that

Ext1R(KX(i), θX(j)

) � Ext1BX(eQX

(KX(i)

), BXΔ(j)

). �

Proposition 4.2. Let X ∈ indF(θ) be θ -directing and such that Ext2R(Q,YX) = 0. Then, for anyM ∈ F(θX), we have the following equalities, where M ′ := eQX(M)

qθ (dim θ M) = qBX Δ(dimBX Δ M′) = χ

BX(dimM ′) and qθ (dim θ X) = 1.

Proof. Since X′ := eQX(X) is BXΔ-directing and BXΔ-omnipresent, we have from 3.13 thatpdF(BXΔ) � 2 and qBX Δ(dimBX Δ X

′) = 1. Therefore, by 4.1, we get the following equalities

qθ (dim θ M) = qBX Δ(dimBX Δ M′) = χ

BXΔ(dimBX Δ

M ′).

On the other hand, from 2.8, we obtain that

χBX

Δ(dimBX ΔM ′) = χBX(dimM ′),

proving the result. �Lemma 4.3. Let X,Z ∈ indF(θ) and Ext2R(Q,YX) = 0. If X is θ -directing and dim θ X =dim θ Z, then X � Z.

Proof. Assume that X is θ -directing and dim θ X = dim θ Z. Let X′ := eQX(X) and Z′ :=eQX(Z). Thus, X

′ is BXΔ-directing and BXΔ-omnipresent. Then, by 3.13, we have that pdX′ � 1and idF(BX Δ) X

′ � 1. On the other hand, we have that dimBX

Δ X′ = dim

BXΔ Z

′ since the functoreQX = HomR(QX,−) :F(θX) → F(BXΔ) is an equivalence of exact categories. Therefore, by2.8 and 4.2, we obtain the following equalities

1 = χB Δ(dim Δ X

′) = 〈dim Δ X′,dim Δ Z′〉B Δ = 〈dimX′,dimZ′〉BX .

X BX BX BX X


Hence 1 = 〈dimX′,dimZ′〉BX . Moreover, applying 1.5 to this equality and using that pdX′ � 1,we get the following equality

1 = dimk HomBX(X′,Z′) − dimk Ext1BX(X′,Z′),

proving that HomBX(X′,Z′) = 0. Similarly, by using that idF(BX Δ) X′ � 1, we obtain that

HomBX(Z′,X′) = 0. Therefore, in view of the BXΔ-directness of X′, we get that X′ � Z′. Then

X � Z, proving the result. �Corollary 4.4. Let (θ,�) be a stratifying system such that Ext2R(Q,YX) = 0 for any X ∈indF(θ). If F(θ) is θ -directing, then the correspondence

X �→ dim θ X

induces an injection from indF(θ) to the set of positive roots of qθ .

Proof. Assume that F(θ) is θ -directing. Then, by 4.2, we know that dim θ X is a positive roofof qθ for any X ∈ indF(θ). Then, the result follows from 4.3. �Lemma 4.5. Let R be an algebra such that the Cartan matrix CR of R is invertible, and (θ,�)be a stratifying system. If pdF(θ) θ � 2 and pd θ < ∞, then for any M ∈ F(θ), we have thefollowing equality

χR(dimM) = dimk End(RM) − dimk Ext1R(M,M) + dimk Ext2R(M,M).

Proof. Assume that pdF(θ) θ � 2 and pd θ < ∞. Since pdF(θ) θ � 2, we get from 3.10(a) thatExtiR(M,M) = 0 for i � 3. On the other hand, the fact that pd θ < ∞ implies that the projectivedimension of F(θ) is finite. Then, the result follows from 1.5. �Lemma 4.6. Let (θ,�) be a stratifying system of size t , and F(θ) be θ -directing. Then, for anynon-negative z ∈ Zt \ {0}, there exists M ∈F(θ) such that dim θ M = z and Ext1R(M,M) = 0.

Proof. Let z ∈ Zt \ {0} be non-negative, so we have that z = (z1, . . . , zt ), where zi � 0 for anyi and zio > 0 for some index io. It is clear that there exists M ∈ F(θ) with dim θ M = z (forexample M = ⊕ti=1 θ(i)zi ). We choose such an M with dimk End(RM) smallest possible. LetM = ⊕si=1 Msii with Mi indecomposable and pairwise non-isomorphic for all i. We assert thatExt1R(Mi,Mj ) = 0 for all i = j . Indeed, suppose that Ext1R(Mi,Mj ) = 0 for some i = j . Thenwe have a non-split exact sequence in F(θ)

0 → Mj → E → Mi → 0.

This sequence gives rise to the following non-split exact sequence

0 →( ⊕

M

)⊕ Mj →

( ⊕M

)⊕ E → Mi → 0.

=i,j =i,j


Let Z := (⊕ =i,j M) ⊕ E. Since F(θ) is closed under direct summands and under extensions,we have that Z ∈ F(θ) and so dim θ Z = z. Then, by Lemma 1 in Section 2.3 in [12], we havethat dimk End(RZ) < dimk End(RM), which contradicts the fact that dimk End(RM) is smallestpossible. Hence Ext1R(Mi,Mj ) = 0 for all i = j . On the other hand, Ext1R(Mi,Mi) = 0 for all isince Mi is θ -directing (see 3.3). �

The following result has an analogue for mod R, see 2.4(9) in [12], and the proof uses theresults given before. As an application, on the one hand, we get the generalization of Theorem 2.7in [2] for standardly stratified algebras; and on the other hand, the generalization of 2.4(9′) in [12]for the category F(θ).

Theorem 4.7. Let R be an algebra, and (θ,�) be a stratifying system of size t such that thefollowing conditions hold:

(a) Ext2R(Q,YX) = 0 for any X ∈ indF(θ),(b) the category F(θ) is θ -directing,(c) pdF(θ) θ � 2 and pd θ < ∞,(d) the Cartan matrix CR is invertible and χθ (dim θ X) = χR(dimX) for any X ∈F(θ).

Then, the quadratic form qθ is weakly positive, and the correspondence

X �→ dim θ Xinduces a bijection from indF(θ) to the set of positive roots of qθ .

Proof. Since (c) and (d) holds, we can use 4.5 to obtain the following equality for any M ∈F(θ)qθ (dim θ M) = dimk End(RM) − dimk Ext1R(M,M) + dimk Ext2R(M,M). (9)

We start by proving that qθ is weakly positive. Indeed, let z ∈ Zt \ {0} be non-negative. Hence,by 4.6, there exists M ∈F(θ) such that dim θ M = z and Ext1R(M,M) = 0. Then, by (9), we ob-tain qθ (z) = qθ (dim θ M) = dimk End(RM) + dimk Ext2R(M,M) > 0, proving that qθ is weaklypositive.

On the other hand, from (a), (b) and 4.4, we conclude that the correspondence X �→ dim θ Xfrom indF(θ) to the set of positive roots of qθ is injective. Moreover, we assert that it is alsosurjective. Indeed, let z be a positive root of qθ . Then by, 4.6 and (9), we get some X ∈ F(θ) suchthat z = dim θ X and 1 = qθ (z) = dimk End(RX)+ dimk Ext2R(X,X). Hence dimk End(RX) = 1,and so X ∈ indF(θ), proving the result. �

The following two results generalize for standardly stratified algebras Theorem 2.7 in [2],which was proved for quasi-hereditary algebras.

Corollary 4.8. Let R be a standardly stratified algebra, and F(RΔ) be RΔ-directing. Ifpd RΔ � 2, then the quadratic form qRΔ is weakly positive, and the correspondence

X �→ dimRΔ

X

induces a bijection from indF(RΔ) to the set of positive roots of qRΔ.


Proof. In order to apply the theorem above, we just have to check that the hypothesis (a) and(d) needed in 4.7 hold. Let (RΔ,Q,�) be the epss associated to (RΔ,�). Since R is a stan-dardly stratified algebra, we get that Q = RR. Therefore, the hypothesis (a) holds. Finally,by 2.8, we get that the Cartan matrix CR is invertible and χRΔ(dimRΔ X) = χR(dimX) forany X ∈ F(RΔ). �Theorem 4.9. Let R be an algebra, and (θ,�) be a stratifying system. If F(θ) is θ -directingthen indF(θ) is finite.

Proof. Suppose that (θ,�) has size t , and assume that F(θ) is θ -directing. For any X ∈indF(θ), we consider the induced stratifying system (θX,�) as before. We assert that indF(θX)is finite for any X ∈ indF(θ). Indeed, since eQX :F(θX) → F(BXΔ) is an equivalence as ex-act categories, it is enough to prove that indF(BXΔ) is finite. Let X′ := eQX (X). Since X isθ -directing, we have that X′ is BXΔ-directing and BXΔ-omnipresent. Then, by 3.13, we havethat pd BXΔ � 2. Therefore, we can apply 4.8 and the well-known fact: “the set of positive rootsof a given weakly positive quadratic form is finite” to get that indF(BXΔ) is finite.

Consider the function Φ : indF(θ) → 2[1,t], where Φ(X) := Suppθ (X). That Φ inducesan equivalence relation ∼ on the set indF(θ), and so this set can be partitioned into classes[X] = Φ−1(Φ(X)) with X ∈ indF(θ). The cardinal number card([X]) of each class [X] is finitesince [X] ⊆ indF(θX). On the other hand, card(indF(θ)/∼) = card(ImΦ) � 2t . Thus, we haveproven that indF(θ) is finite. �

The following example shows that the converse of the above theorem does not hold.

Example 4.10. Let R be the quasi-hereditary algebra, which has the following presentation

modulo the ideal I , of the path algebra, generated by the paths α2α1, β1β2, α2β2, and α1β1 −β2α2. Then indF(RΔ) is finite but not directing, see [17].

5. Examples of θ -directing

In this section, we give examples of algebras R such that mod R is not directing, but thecategory F(θ) is θ -directing. Therefore, we get by 4.9 that F(θ) is of finite representation type.Observe, that R is not so, since mod R is not directing.

Let R be an algebra, and let C be a component of the Auslander–Reiten quiver ΓR of mod R.We say that C is a directing component if addC is directing (see at the beginning of Section 3);that is, any X ∈ C is directing in mod R. Note that a cyclic path in ΓR gives raise to a cyclein mod R. However, there usually exists cycles in mod R consisting of modules belonging todifferent components of ΓR .

Proposition 5.1. Let R be an algebra and (θ,�) be a stratifying system. If θ is contained into adirecting component C of ΓR and addC is closed under extensions, then indF(θ) is finite.


Proof. Assume that θ is contained into a directing component C of ΓR and addC is closed underextensions. Due to 4.9, we need to prove that F(θ) is θ -directing. To do that, it is enough to seethat F(θ) ⊆ addC since, by hypothesis, we know that C is directing. Indeed, let 0 = M ∈ F(θ).We apply induction on θ (M) := ∑i[M : θ(i)]. If θ (M) = 1 then M ∈ add θ(i) for some i, andso M ∈ addC since θ ⊆ C.

Suppose that θ (M) > 1 and that X ∈ addC if θ (X) < θ (M). Then, by Proposition 2.9in [14], we have an exact sequence 0 → N → M → θ(i)mi → 0 in F(θ) such that θ (N) <

θ (M). Hence, N ∈ addC. Therefore M ∈ addC since the category addC is closed under exten-sions, proving that F(θ) ⊆ addC. This yields that F(θ) is directing, and so F(θ) has to be, inparticular, θ -directing. �

According to 5.1, we need to find a directing component C of ΓR such that addC be closedunder extensions. As we will see, the preprojective component is an example of such kind ofcomponents. Following C.M. Ringel in [12, p. 80], we recall that C is a preprojective componentif C, as a translation quiver, is preprojective. That means that C is a translation quiver withoutcyclic paths, with only finitely many τ -orbits and such that any τ -orbit contains a projectivevertex. We also recall that a component C is closed under predecessors if for any indecomposableR-modules M and N such that N ∈ C and HomR(M,N) = 0 we have that M ∈ C. The followingis one of the main properties of a preprojective component (see [12, p. 80]).

Proposition 5.2. [12] Let R be an algebra, and C be a preprojective component of ΓR . Then, Cis a directing component which is standard and closed under predecessors.

Lemma 5.3. Let R be an algebra and C be a component of ΓR . If C is closed under predecessorsthen addC is closed under submodules and extensions.

Proof. Suppose that the component C is closed under predecessors. We start by proving that Cis closed under submodules. Indeed, Let M ∈ addC and N be a submodule of M . We have adecomposition N = ∐i Ni and M = ∐j Mj into indecomposable modules. For any i, there issome index j0 such that the composition of the projection πj0 :

∐j Mj → Mj0 and the inclusion

ıi :Ni → ∐j Mj is non-zero. Hence, Ni ∈ C since C is closed under predecessors, proving thatC is closed under submodules.

Finally, we prove that C is closed under extensions. Let 0 → M → E f−→ N → 0 be an exactsequence with M and N belonging to addC. Then, there is a decomposition E = X ∐Y such thatf |Y = 0 and g := f |X :X → N is right minimal. On the one hand, Y is a submodule of M , and soY ∈ addC since it is closed under submodules. On the other hand, we assert that X also belongsto addC. Indeed, let X′ be a indecomposable direct summand of X. Since g :X → N is rightminimal, we get that g|X′ = 0. Therefore, there is some indecomposable direct summand N ′ ofN such that HomR(X′,N ′) = 0. Then we have that X′ ∈ C since C is closed under predecessors,proving that E ∈ addC. �Theorem 5.4. Let R be an algebra and (θ,�) be a stratifying system. If θ is contained into apreprojective component of ΓR , then indF(θ) is finite.

Proof. Suppose that θ is contained into a preprojective component C of ΓR . Then, by 5.2 and 5.3,we get that C is directing and addC is closed under extensions. Therefore, the result followsfrom 5.1. �


Now, we give examples of stratifying systems belonging to a preprojective component. Inorder to do that, we assume that R is a connected hereditary algebra of infinite representationtype. In that case, it is well known that R is the path k-algebra kQ, where Q is a connected,without cycles and not Dynkin quiver. Since the quiver Q has no cycles, we can label the setof vertices Q0 = {1,2, . . . , s} in such a way that the standard module RΔ(i) is the projectiveR-module P(i) associated to the vertex i. So we can see, in this way, the hereditary algebra R asa quasi-hereditary algebra (R,�), where � is the natural total order on Q0. We recall now somewell-known facts about hereditary algebras (see, for example, in [12]). Since R is connected, wehave only one preprojective component of ΓR and will be denoted by P .

Next, we recall the structure of the preprojective component P . Let Qop be the opposite quiverof Q and ZQop the translation quiver associated to Qop. We denote by NQop the sub-translationquiver of ZQop with vertices (n, i) such that n � 0 and i ∈ Q0. Identifying the complete setof projective modules {P(i): i ∈ Q0} with the set of vertices Q0, we get that P = NQopsince R is of infinite representation type. Informally speaking, the set of projective modules{P(i): i ∈ Q0} becomes into a “sectional path” P0 in P and the preprojective component can beseen as

⋃n�0 τ

−nP0, where τ is the Auslander–Reiten translation. Then, we set θn := τ−nP0.That is θn(i) := τ−nP (i) for i ∈ Q0. It is easy to see, that θ0 = P0 = RΔ.

We assert that, for any natural number n, the pair (θn,�) is a stratifying system of size s andF(θn) = add θn. Indeed, on the one hand, we have HomR(θn(j), θn(i)) � HomR(RΔ(j), RΔ(i)).On the other hand, by using the Auslander–Reiten’s formula, we get Ext1R(θn(j), θn(i)) �D HomR(θn(i), τθn(j)) = 0 for any i, j ∈ Q0, proving the statement.

Acknowledgment

The authors thank Professor M.I. Platzeck for helpful discussions.

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